My Example:
Players 1 through n are playing a game and in each round there is one winner and after the winner is determined, one of the non-winners gets randomly taken out of the total players pool (i.e. round 1 = n total players, round 2 = n-1, round 3 = n-2, etc).
I am trying to show that if we look at a large sample of these games, a strong player (potentially the winner) should typically have a higher amount of competition round wins at the end of each game. However, I do not understand how to go about adjusting for the increased probability that a player might win each round when a non-winner is eliminated [i.e. round 1 = $\frac{1}{n}$ % chance of winning that round, round 2 = $\frac{1}{n-1}$, round 3 = $\frac{1}{n-2}$, etc]. This is assuming of course that each player has an equal chance of winning each round, which is obviously what i'm trying to get around.
Does anyone know how I might account for this increased probability and possibly how to also take into consideration as the game goes on that a stronger player who has won in the first round (or other future rounds) is given a higher probability than the rest to win the next rounds?
Basically, how do I calculate the strength of a player and add it into a predictive model on how they will do based on past games with other players data.
Thank you for any help in advance.
There are multiple ways to solve this, but you can give each player the following score :
$\mathbf{S}(x) = \underset{\{i\ :\ \text{x won round i}\}}{\sum \frac{1}{p_i}}=\underset{\{i\ :\ \text{x won round i}\}}{\sum (n- i)}$
With $p_i = \frac{1}{n -i}$ being the probability of winning the round $i$. So the earlier rounds count for more than the later ones.
If you want the effect of the round won to be smaller, you can take
$\mathbf{S}(x) = \underset{\{i\ :\ \text{x won round i}\}}{\sum 1 + \frac{1}{k}\frac{1}{p_i}}= \text{number of wins} + \underset{\{i\ :\ \text{x won round i}\}}{\sum \frac{1}{k}(n- i)}$
So you are still counting all the wins of the player, and then adding extra points for winning in the earlier rounds.